Real nonparametric regression using complex wavelets

Summary.  Wavelet shrinkage is an effective nonparametric regression technique, especially when the underlying curve has irregular features such as spikes or discontinuities. The basic idea is simple: take the discrete wavelet transform of data consisting of a signal corrupted by noise; shrink or remove the wavelet coefficients to remove the noise; then invert the discrete wavelet transform to form an estimate of the true underlying curve. Various researchers have proposed increasingly sophisticated methods of doing this by using real-valued wavelets. Complex-valued wavelets exist but are rarely used. We propose two new complex-valued wavelet shrinkage techniques: one based on multiwavelet style shrinkage and the other using Bayesian methods. Extensive simulations show that our methods almost always give significantly more accurate estimates than methods based on real-valued wavelets. Further, our multiwavelet style shrinkage method is both simpler and dramatically faster than its competitors. To understand the excellent performance of this method we present a new risk bound on its hard thresholded coefficients.     

Strong convergence rates of probabilistic integrators for ordinary differential equations

Probabilistic integration of a continuous dynamical system is a way of systematically introducing discretisation error, at scales no larger than errors introduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al. (Stat Comput 27(4):1065–1082, 2017. https://doi.org/10.1007/s11222-016-9671-0), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator. These and similar results are proven for probabilistic integrators where the random perturbations may be state-dependent, non-Gaussian, or non-centred random variables.

     

Using the EM-algorithm for survival data with incomplete categorical covariates

Incomplete covariate data is a common occurrence in many studies in which the outcome is survival time. With generalized linear models, when the missing covariates are categorical, a useful technique for obtaining parameter estimates is the EM by the method of weights proposed in Ibrahim (1990). In this article, we extend the EM by the method of weights to survival outcomes whose distributions may not fall in the class of generalized linear models. This method requires the estimation of the parameters of the distribution of the covariates. We present a clinical trials example with five covariates, four of which have some missing values.     

THE FAMILY CRISIS INTERVENTION LITERATURE: WHAT IS MEANT BY “FAMILY”?*

Family crisis intervention is a rapidly growing area of clinical care in family therapy which lacks clear conceptualization, especially with respect to how “family” fits into crisis intervention theory. This paper integrates recent concepts from the family therapy literature into three views: family as background, family as context, and family as a unit. The family crisis intervention literature (56 publications) is reviewed and categorized according to these three views. Clarity in how “family” is viewed and consistency between theory and practice are examined. Recommendations for improving the conceptualization of “family” in family crisis intervention theory and practice are made.     

Looking Forward Not Looking Back: The Evaluation of Community Living

This article outlines the development of evaluative methods within the field of mental handicap services. It suggests that evaluators have too often taken for granted the tacit assumptions underlying service models, using these to determine the criteria for evaluating success. It argues that evaluators should be more explicit about the values they adopt, and what patterns of life services should be enabling people to achieve. By more closely linking evaluation criteria to what people with a mental handicap should achieve, evaluation can become more effective in promoting positive change in service models. The article describes an approach developed by the authors for the evaluation of the 'All-Wales Strategy for the Development of Services for Mentally Handicapped People'.     

An analysis of two probabilistic model induction techniques

Two probabilistic model induction techniques, cart and constructor, are compared, via a series of experiments, in terms of their ability to induce models that are both interpretable and predictive. The experiments show that, although both algorithms are able to deliver classifiers with predictive performance close to that of the optimal Bayes rule,constructor is able to generate a probabilistic model that is more easily interpretable than the cart model. On the other hand, cart is a more mature algorithm and is capable of handling many more situations (e.g., real-valued training sets) thanconstructor. A variety of characteristics of both algorithms are compared, and suggestions for future research are made.     

Participation in Practice

This paper is one of a series written by members of the Mental Handicap Research Project at the University College of North Wales in connection with the All-Wales Strategy. Previous papers describe local arrangements for the planning of new services and discuss in general terms issues concerning partnership and participation. This paper describes modes of parental participation in the planning of new services for the mentally handicapped.